Systems of Linear Equations • Topic 2 of 4

Substitution Method

Substitution works well when one equation already gives a variable alone (or is easy to rearrange). Solve one equation for a variable, substitute that expression into the other equation to get a single-variable equation, solve it, then back-substitute to find the second variable. For y = 2x and x + y = 9, replace y to get x + 2x = 9, so x = 3 and y = 6. Always substitute into the other equation, not the one you solved. Substitution is reliable on the SAT whenever a variable is, or can be, isolated cheaply.

✅ Solved examples

1. Solve y = 2x and x + y = 9.
Substitute: x + 2x = 9 → 3x = 9 → x = 3, then y = 6.
2. Solve x = y + 1 and 2x + y = 8.
Substitute: 2(y + 1) + y = 8 → 3y + 2 = 8 → y = 2, x = 3.
3. Solve y = 3x − 1 and y = x + 5.
Set equal: 3x − 1 = x + 5 → 2x = 6 → x = 3, y = 8.
4. Solve y = x and 2x + 3y = 10.
Substitute: 2x + 3x = 10 → 5x = 10 → x = 2, y = 2.

✏️ Practice — try these, take hints as needed

1. Solve y = 3x and x + y = 8.
Substitute 3x for y.
x + 3x = 8.
Solve for x then y.
x = 2, y = 6.
2. Solve x = y − 2 and x + y = 10.
Substitute (y − 2) for x.
(y − 2) + y = 10.
Solve.
y = 6, x = 4.
3. Solve y = 2x + 1 and y = x + 4.
Set the two expressions equal.
2x + 1 = x + 4.
x = 3.
x = 3, y = 7.
4. Solve y = 4x and 2x + y = 18.
Substitute 4x for y.
2x + 4x = 18.
Solve.
x = 3, y = 12.
5. Solve x = 2y and x + 3y = 20.
Substitute 2y for x.
2y + 3y = 20.
Solve for y then x.
y = 4, x = 8.

📝 Topic test — 8 questions

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